1. Theo's mistake is in the second step where he multiplied the top of the expression on the left by (x+2). First of all it is not necessary to do that multiplication because the expression at the bottom (x-5)(x+2) is already the common denominator and therefore there is no need to multiply this expression by any factor. Secondly, any time you multiply the top you should also multiply the bottom to preserve the ratio. Theo only multiplies the top and not the bottom and therefore he has altered the ratio. Paula's mistake is in the fourth step where she factors x^2-4x-5 into (x+2)(x-5) which is wrong factoring. Also, factoring is not needed here. Since the denominators of the two expressions is the same the numerator can be added. The mistake continues in the next step where she cancels out factors across an addition. You cannot do that. There is a plus sign between the two expressions and common factors cannot be cancelled out across addition or subtraction.

2. Least Common Denominator or LCD is the least common multiple of the denominators of two or more fractions. It is the smallest denominator that two or more fractions have in common. To find LCD do the following: 1) Factor each of the denominators. 2) Find the least common multiple among all the denominators. If the same factor occurs in all denominators, then the factor will occur only once in LCD. If a factor occurs in one denominator and not the other, then that factor has to go into the LCD. Once you find all the factors that go into the LCD, the product of the factors will be the LCD. 3) Once you find the LCD, make each fraction have the LCD in the denominator. You can do that by multiplying the numerator and denominator of each fraction by the same factor to make the denominator the same as LCD. 4) Once the LCD is the same, the numerator of each fractions can be added or subtracted and simplified.